Consider the linear function that is represented by the equation y = 4 x + 5 and the linear function that is represented by the table below.

x
y
2
16
4
26
6
36
8
46

Which statement is correct regarding their slopes and y-intercepts?
The function that is represented by the equation has a steeper slope and a greater y-intercept.
The function that is represented by the equation has a steeper slope, and the function that is represented by the table has a greater y-intercept.
The function that is represented by the table has a steeper slope, and the function that is represented by the equation has a greater y-intercept.
The function that is represented by the table has a steeper slope and a greater y-intercept.

1 answer

To compare the slopes and y-intercepts of the two linear functions, we start by analyzing the given equation y = 4x + 5:

  1. Slope and Y-Intercept of the Equation:
    • The slope (m) of the equation y = 4x + 5 is 4.
    • The y-intercept (b) is 5.

Now, we'll determine the slope and y-intercept of the function represented by the table:

Data from the table:

  • When x = 2, y = 16
  • When x = 4, y = 26
  • When x = 6, y = 36
  • When x = 8, y = 46

Calculating the slope of the table:

Using two points from the table, we can calculate the slope. Let's take points (2, 16) and (4, 26):

\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{26 - 16}{4 - 2} = \frac{10}{2} = 5 \]

Y-Intercept of the Table Function:

To find the y-intercept, we can use the slope we calculated (5) and one of the points. We'll use the point (2, 16):

We know that: \[ y = mx + b \implies 16 = 5(2) + b \] \[ 16 = 10 + b \implies b = 6 \]

Summary of both functions:

  1. For the function represented by the equation y = 4x + 5:

    • Slope: 4
    • Y-Intercept: 5
  2. For the function represented by the table:

    • Slope: 5
    • Y-Intercept: 6

Comparison:

  • The slope of the function from the table (5) is steeper than the slope of the equation (4).
  • The y-intercept of the function from the table (6) is greater than the y-intercept of the equation (5).

Conclusion:

The correct statement is: The function that is represented by the table has a steeper slope and a greater y-intercept.

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